Devin attempted to solve the inequality -2x+6>10 following the steps below:
•subtract 6 from both sides of the inequality.
•divide both sides of the inequality by -2.
He finds that x>-2. Clearly explain why this is not a correct solution. In your response, be sure to explain what Devin needs to do to find the correct solution.
multiplying or dividing by negative values changes the direction of the inequality. So, the answer is x < -2
To see why this is so, work the problem without multiplying by negative values. In this case, that can be done as follows:
-2x + 6 > 10
subtract 6
-2x > 4
add 2x
0 > 4 + 2x
subtract 4
-4 > 2x
-2 > x
see, multiplying by a negative is like swapping things to the other side and multiplying by a positive. So, the effective direction of the inequality changes.
To explain why Devin's solution is incorrect, let's go through the steps he followed and understand the mistake he made:
Step 1: Subtract 6 from both sides of the inequality.
-2x + 6 - 6 > 10 - 6
-2x > 4
So far, so good. Devin correctly subtracted 6 from both sides of the inequality.
Step 2: Divide both sides of the inequality by -2.
(-2x) / -2 > 4 / -2
x > -2
Here's where Devin went wrong. He divided both sides of the inequality by -2, which is the same as multiplying by -1/2. When we multiply or divide an inequality by a negative number, we need to reverse the direction of the inequality symbol.
In other words, when we multiply or divide both sides of an inequality by a negative number, the inequality sign changes direction. So, instead of x > -2, the correct solution should be x < -2.
To find the correct solution, Devin needs to reverse the inequality sign when dividing both sides by -2. The correct steps would be:
1. Subtract 6 from both sides of the inequality:
-2x + 6 - 6 > 10 - 6
-2x > 4
2. Divide both sides of the inequality by -2, remembering to reverse the inequality sign:
(-2x) / (-2) < 4 / (-2)
x < -2
So, the correct solution to the inequality -2x + 6 > 10 is x < -2.